He added that he believes Trump's public criticism of Mueller is "unhelpful" and agrees with his colleagues who have said firing the special counsel "would be a mistake and produce all sorts of unintended consequences."

Cornyn declined to say what the potential consequences would be.

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The latest round of speculation about Mueller's future comes after Trump teed off over the weekend against the probe into Russia's election meddling and potential ties between the Trump campaign and Moscow, saying the probe is made up of "hardened Democrats."

"A total WITCH HUNT with massive conflicts of interest!" Trump tweeted on Monday.

The White House has stressed that there are no plans to fire Mueller, though The New York Times reported earlier this year that Trump ordered his staff to fire Mueller last year but ultimately backed down when his White House counsel threatened to resign.

And while this weekend's tweets sparked a new rush from lawmakers to defend Mueller, GOP senators also appeared deeply skeptical on Monday that legislation is needed or that Trump would actually fire the special prosecutor, who is widely respected in Washington.

"If you're going to pick a fight, this is the fight to pick. You've got to pick this fight. If you don't pick this fight, we might as well not be here. This is a serious one. ... The best thing would be — is, preemptively to convince the president, 'don't go there,'" he said.

Corker later clarified that he wasn't saying he would support such a move — and noted he wasn't voting for the mammoth government funding bill, regardless — but "I just suggested that if I were the Senate Democratic leadership I would be pushing for that. ... I just assumed that Sen. [Charles] Schumer [N.Y.] and ... Leader [Nancy] Pelosi [Calif.] would be pushing for that."